If your definition of pure math is "proving theorems", she can work in the following (math related to the following):

1). Control theory (this can mean doing dynamical systems, PDEs, etc): PhDs hired by several leading companies such as Ford, Bosch, Honda, United Technologies

2). Information theory and networks : Internet companies

3). Optimization and/or nonlinear programming: PhDs hired by numerous companies

4). Communication theory: PhDs hired by communication giants

5). Stochastic geometry: Same as 4

6). Probability: Finance/banks/internet